Finding Cartesian from Parametric Equations

Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.

x = 1 + sin t, y = cos t − 2, 0 ≤ t ≤ π

Verified step by step guidance

Start with the given parametric equations: \(x = 1 + \sin t\) and \(y = \cos t - 2\), where \(0 \leq t \leq \pi\).

Isolate the trigonometric functions from the parametric equations: from \(x = 1 + \sin t\), we get \(\sin t = x - 1\); from \(y = \cos t - 2\), we get \(\cos t = y + 2\).

Recall the Pythagorean identity for sine and cosine: \(\sin^2 t + \cos^2 t = 1\).

Substitute the expressions for \(\sin t\) and \(\cos t\) into the identity: \((x - 1)^2 + (y + 2)^2 = 1\).

This equation represents the Cartesian form of the particle's path, which is a circle centered at \((1, -2)\) with radius 1. The parameter interval \(0 \leq t \leq \pi\) indicates the particle moves along the upper half of this circle, starting at \(t=0\) and ending at \(t=\pi\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex paths and motions in the plane.

Eliminating the Parameter to Find Cartesian Equations

To find a Cartesian equation from parametric equations, the parameter t is eliminated by using algebraic manipulation or trigonometric identities. This process converts the parametric form into a single equation relating x and y, describing the particle’s path in the xy-plane.

Graphing and Interpreting Parametric Curves

Graphing parametric curves involves plotting points (x(t), y(t)) over the given parameter interval. Understanding the direction of motion and the portion of the curve traced requires analyzing how x and y change as t increases, which helps visualize the particle’s trajectory and orientation.

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Textbook Question

Implicitly Defined Parametrizations

Assuming that the equations in Exercises 15−20 define x and y implicitly as differentiable functions x=f(t), y=g(t), find the slope of the curve x=f(t), y=g(t) at the given value of t.

x sin t + 2x = t, t sin t − 2t = y, t = π

Textbook Question

Polar to Cartesian Equations

Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.

r² = 4r sin θ

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Textbook Question

Finding Parametric Equations

In Exercises 31–36, find a parametrization for the curve.

the ray (half line) with initial point (-1,2) that passes through the point (0,0)

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Textbook Question